L11a271

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_4,9,5,10 X_14,5,15,6 X_18,8,19,7 X_22,15,9,16 X_20,17,21,18 X_16,21,17,22 X_6,20,7,19 X_8,13,1,14
Gauss code	{1, -2, 3, -4, 5, -10, 6, -11}, {4, -1, 2, -3, 11, -5, 7, -9, 8, -6, 10, -8, 9, -7}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{3}v^{2}-2u^{3}v+2u^{2}v^{3}-7u^{2}v^{2}+6u^{2}v-2u^{2}-2uv^{3}+6uv^{2}-7uv+2u-2v^{2}+2v}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {13}{q^{9/2}}}-{\frac {14}{q^{11/2}}}+{\frac {11}{q^{13/2}}}-{\frac {9}{q^{15/2}}}+{\frac {6}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-a^{9}z+a^{7}z^{5}+a^{7}z^{3}-a^{7}z+2a^{5}z^{5}+5a^{5}z^{3}+4a^{5}z+a^{5}z^{-1}+a^{3}z^{5}+a^{3}z^{3}-2a^{3}z-a^{3}z^{-1}-az^{3}-2az$ (db)
Kauffman polynomial	$a^{12}z^{6}-3a^{12}z^{4}+2a^{12}z^{2}+3a^{11}z^{7}-9a^{11}z^{5}+6a^{11}z^{3}-a^{11}z+4a^{10}z^{8}-11a^{10}z^{6}+7a^{10}z^{4}-2a^{10}z^{2}+3a^{9}z^{9}-6a^{9}z^{7}+3a^{9}z^{5}-4a^{9}z^{3}+2a^{9}z+a^{8}z^{10}+3a^{8}z^{8}-11a^{8}z^{6}+9a^{8}z^{4}-3a^{8}z^{2}+5a^{7}z^{9}-12a^{7}z^{7}+13a^{7}z^{5}-6a^{7}z^{3}+3a^{7}z+a^{6}z^{10}+2a^{6}z^{8}-6a^{6}z^{6}+8a^{6}z^{4}-2a^{6}z^{2}+2a^{5}z^{9}-6a^{5}z^{5}+12a^{5}z^{3}-6a^{5}z+a^{5}z^{-1}+3a^{4}z^{8}-5a^{4}z^{6}+5a^{4}z^{4}-2a^{4}z^{2}-a^{4}+3a^{3}z^{7}-6a^{3}z^{5}+5a^{3}z^{3}-4a^{3}z+a^{3}z^{-1}+2a^{2}z^{6}-4a^{2}z^{4}+a^{2}z^{2}+az^{5}-3az^{3}+2az$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

1

-1

-2

4

1

3

-4

5

2

-3

-6

6

3

-8

7

5

-2

-10

7

6

1

-12

5

8

3

-14

4

6

-2

-16

2

5

3

-18

1

4

-3

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$